Cálculo numérico da forma normal de Floquet e aplicações em controle de sistemas dinâmicos

Abstract

This work presents a rigorous numerical method to compute the Floquet Normal Form X(t) = Q(t)etR for a given Fundamental Solution of a _??periodic Linear Di_erential Equation. This problem is replaced by solving an equation f(x) = 0 such that f : s ! s, is de_ned in a suitable Banach Space s. The Method aims to _nd an approximate solution _x to the equation f(x) = 0 and r > 0 such that there exists a point x_ 2 B(_x;01)(r) _ s, f(x_) = 0. The technique is based on the de_nition of an operator T : s ! s whose _xed points are solutions of the equation f(x) = 0. Thus, the numerical technique allows to calculate r so that T : B(_x;01)(r) ! B(_x;01)(r) is a contraction, providing conditions to applicate the Banach Fixed Point Theorem to ensure the existence of a _xed point of the operator T, and hence the existence of a solution to the equation f(x) = 0 in B(_x;01)(r). Radii Polinomials, presented in [11], [27] and [5], are used to compute r. In order to prevent a loss of accuracy due to rounding errors, we use interval arithmetic for performing the calculations. A MATLABR 2008 code is used to implement the numerical method described. It's also addressed the application of the Floquet Normal Form in the control of two classical dynamic systems, Forced Pendulum and Du_ng Oscillator. The goal is that the orbits of these systems become asymptotically close to a desired trajectory y : R ! R, lim t!1[x (t) ?? y (t)] = 0. This problem is often replaced in Electrical Engineering by its linearized version, which leads to the use of Floquet Normal Form. Chapter 1 provides an introduction to the Floquet theory and other results used in the rest of the work. Chapter 2 contains a description of the numerical method for calculating the Floquet normal form and its proof. Chapter 3 provides details of the method as a computational algorithm. Chapter 4 contains the application of Floquet normal form to the control of the Forced Pendulum and the Du_ng Oscillator, as previously mentioned. Appendices present concepts and auxiliary statements used throughout the work.Key Words: Floquet Theory, Periodic Linear Systems, Rigorous Numerical Method, Radii Polinomials, Control.